Conditional Stable Matchings

In matching theory of contracts the substitutes condition plays an essential role to ensure the existence of stable matchings. We study many-to-many matchings where groups of individuals, of size possibly greater than two, are matched to a set of institutions. Real-world examples include orphan brothers accepting an adoptive family conditional on all of them being included; hiring contracts that may only be chosen together; or a situation where a firm accepts to hire several workers only if they accept to work on different days (part-time jobs). We demonstrate by several examples that such extra conditions may alter the natural choice maps so that stable matchings cannot be obtained by applying the standard theorems. We overcome this difficulty by introducing a new construction of choice maps. We prove that they yield stable matchings if the construction respects an "anti-trust" rule on the supply side of the market.